I was excited to find that the semidihedral group of order 16 matches what I'm looking for as a model. The larger order semidihedral groups unfortunately do not generalize the relations I'm looking for:
$$G_{2^{2n}} = \langle a, x \ |\ a^{2^{n+1}}=e\ ,\ x^{2^{n-1}}=e\ ,\ xax^{-1}=a^{3}\rangle$$
Edit: In case that's hard to read supers on supers, I'm looking for groups that are generated by $\langle a,x \rangle$ with $$\left\vert a \right\vert = 2^{n+1}, \left\vert x \right\vert=2^{n-1}, xa=a^3x$$
(with $n\geq 2$). For $n=2$, $G_{16}=SD_{16}$. But for larger $n$ the semidihedral groups do not have the same structure. The key is that the order of $a$ and $x$ should both double for each increment in $n$, but the relation $xa=a^3x$ is the same for all $n$. I'm not sure if the groups are well-studied and well-classified as a family.
I'm not an expert in group theory, but I'd like to study these groups and learn as much as I can about group theory along the way. Thanks in advance for any help you can provide!